Math 132, Spring 2011 - Exam 3NAME:STUDENT ID NUMBER:This exam contains sixteen questions. The first fourteenare multiple choice questions and count for five pointseach. There is no partial credit on these questions, soread each question carefully, check your arithmeticand make sure that you have marked the answer you in-tended to mark. The last two questions, which are eachworth fifteen p oints, require written answers, and somepartial credit might be given. However, no credit will begiven for information that is not germane to the problemat hand. Please make sure to write your name and stu-dent ID number on the pages that include your answersto the last two questions. In fact, you will get onepoint on each of these two questions for writingyour name and ID number legibly.11. Let an= 3 − (0.4)n. The sequence {an}(a) converges to −1(b) converges to 0(c) converges to 1(d) converges to −3(e) converges to 3(f) diverges to ∞(g) diverges to −∞(h) diverges but not to ∞ or −∞22. Let an= (1 +1n)n. Compute limn→∞an.(a) 0(b) 1(c) e(d) 3(e) π(f) e3(g) π3(h) ∞33. Which of the following sequences are monotonic?(A)an=12n −1(B)an=(−1)nn, (C)an= 2n+3, (D)an= cos(nπ)(a) A only(b) A and B only(c) A and C only(d) A and D only(e) A, B and C only(f) A, B and D only(g) A, C and D only(h) A, B, C and D44. Compute the sum of the series∞Xn=23(−12)n(a) 0(b) 0.5(c) 1.5(d) −1.5(e) 2(f) 6(g) −∞(h) ∞55. Compute the sum of the series∞Xn=12n−1− 35n(a) 0(b) ∞(c) −∞(d) 1/3(e) 1/2(f) 3/5(g) −5/12(h) −1/566. Determine whether the series∞Xn=1(1(n + 1)2−1n2)converges or diverges. If it converges, compute thesum.(a) diverges to ∞(b) diverges to −∞(c) diverges but not to −∞ or ∞(d) −1(e) −2/3(f) 0(g) 1(h) 4/577. Which of the following is the best estimate of theerror R in the approximation∞Xn=11n3≈ s100(a) R ≤ 0.001(b) R ≤ 0.005(c) R ≤ 0.0001(d) R ≤ 0.0005(e) R ≤ 0.00001(f) R ≤ 0.00005(g) R ≤ 0.000001(h) R ≤ 0.00000588. Which of the following three series is convergent?(A)∞Xn=1n + 2n3/2(B)∞Xn=13n3n4− 2n2(C)∞Xn=1sin(n)√n4+ 5(a) A only(b) B only(c) C only(d) A and B only(e) A and C only(f) B and C only(g) all(h) none99. Apply the Ratio Test to∞Xn=1n!nnFind L and, if possible, decide whether the seriesconverges or diverges.(a) L = 0 and the series converges by the Ratio Test.(b) L = 0 and the series diverges by the Ratio Test.(c) L = 1 and the Ratio Test fails.(d) L = 1 and the series converges by the Ratio Test.(e) L = e−1and the series diverges by the Ratio Test.(f) L = e−1and the series converges by the RatioTest(g) L = ∞ and the series diverges by the Ratio Test.(h) L = ∞ and the series converges by the RatioTest.1010. Which of the following three alternating series isconvergent?(A)∞Xn=1(−1)n1n + 2(B)∞Xn=1(−1)nnn4+ 9(C)∞Xn=3(−1)ncos(πn)(a) A only(b) B only(c) C only(d) A and B only(e) A and C only(f) B and C only(g) all(h) none1111. Which of the following three alternating series isabsolutely convergent?(A)∞Xn=1(−1)n1n + 2(B)∞Xn=1(−1)nnn4+ 9(C)∞Xn=3(−1)ncos(πn)(a) A only(b) B only(c) C only(d) A and B only(e) A and C only(f) B and C only(g) all(h) none1212. Find the radius of convergence for the series∞Xn=1(2x)nn!(a) R = 0(b) R = 1(c) R = 1/2(d) R = 2(e) R = n!(f) R = x(g) R = ∞(h) R cannot be determined1313. For what values of x does the series∞Xn=1(3x + 1)n10nconverge absolutely?(a) −1 < x < 1(b) −1 < x < 3(c) 0 < x < 3(d) −3 < x < 3(e) −11/3 < x < 3(f) 0 < x < 10(g) −1/10 < x < 10(h) −8 < x < 121414. For which values of x can the functionf(x) =11 + 2xbe expressed as a power series of the form∞Xn=0(−1)n(2x)n?(a) all x(b) All x except x = 1/2(c) −1 < x < 1(d) −1/2 < x < 1/2(e) −2 < x < 2(f) −9 < x < 9(g) −29< x < 29(h) −92< x < 9215Name: Student ID:15. Approximate the sum of the series∞Xn=1(−1)n+1n10nwith an error less than 0.00005.16Name: Student ID:16. Use the Integral Test to decide whether∞Xn=34n (ln n)2converges or diverges. Show your work. Rememberthat you must show clearly that all conditions of theIntegral Test are satisfied.17Name: Student ID:18Name: Student
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